The incomplete elliptic integrals , , , and can be represented through some related functions, for example:

Relations to inverse functions

The incomplete elliptic integral is related to the Jacobi amplitude by the following formulas, which demonstrate that the Jacobi amplitude is within a restricted domain, the inverse function of elliptic integral :

Representations through other incomplete elliptic integrals

The incomplete elliptic integrals , and can be represented through incomplete elliptic integral by the following formulas: